Gaussian Matrices and Exterior Forms: Deterministic Principal Ideals

Abstract

For an infinite Gaussian matrix with independent entries and doubly monotone variances, we identify its algebraic principal ideal by a deterministic shell profile. The exact boundedness condition also implies compactness. A lower singular-value estimate uniform in the mean gives an exact sum law under each fixed compact perturbation. Conditional products and spatial compression yield the corresponding classification for Gaussian evaluations of Hilbert two-forms. For Wiener area of covariance rank at least two, the generator is the covariance tail divided by its index. The area and covariance generate the same ideal precisely under regularity at infinity.

arXiv comment15 pages